Lab 5 The Green House Effect Pre-lab

Transcription

1 Lab 5 The Green House Effect Pre-lab NAME Partner Given information: Radius of the Sun: X 10 8 m Radius of the Earth: X 10 6 m Emissivity of Sun: ~1.0 Surface temperature of the Sun: 5,778 K Average distance between Sun and Earth: X m Answer the following questions before coming to lab. 1) Use Wien s Law to find the peak wavelength for radiation coming from the sun. 2) Humans can see light with wavelengths between 390nm and 700nm. How does the peak wavelength radiation from the sun compare to this range for visible light? 3) Given the temperature of the sun given above, use the formula for thermal radiation to determine the power output of the sun in Watts. 1

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3 Given information: Radius of the Sun: X 10 8 m Radius of the Earth: X 10 6 m Emissivity of Sun: ~1.0 The Green House Effect The One Layer Model Surface temperature of the Sun: 5,778 K Average distance between Sun and Earth: X m Show any required calculation when answering the questions below. NAME Partner In this handout, we want to walk through the calculations to try predict the temperature of the earth from the simple thermodynamic equations we have learned. Part I Bare Earth Model 1) Use Wien s Law to find the peak wavelength for radiation coming from the sun. 2) Given the temperature of the sun given above, use the formula for thermal radiation to determine the power output of the sun in Watts. 3) The intensity of the sun s radiation is defined as how much power hits a square meter at some particular distance. Using the formula for a sphere given on our formula sheet, calculate the intensity of the sun s radiation on the upper surface of the atmosphere of the earth. This is called the solar constant, S or sometimes the solar irradiance. 3

4 4) Our calculation so far has been based on some known constants and simple geometry. Figure 1 to the right shows the solar irradiance as actually measured by satellites. How does our number compare to the actual data? Figure 1 - Total solar irradiance (power received from the Sun) as observed directly by satellites (Courtesy NASA-GISS) You might notice that the power from the sun goes up and down on roughly a 22 year cycle. This is due to sun spots. We ll come back to this cycle in the homework. 5) The irradiance we just calculated is the amount that hits a square meter of the earth s upper atmosphere when it strikes perpendicular. Since the earth is a sphere and not a flat disk, it only strikes this way close to the equator while almost none strikes a square meter near the poles. The average irradiance is ¼ of that. (This is the difference between the area of a disk ( r 2 ) and the area of a sphere (4 r 2 ). Divide your number from question (3) by 4. (In figure 2 below, this irradiance is shown as R TOTAL.) 6) Approximately 31% of this radiation is reflected by clouds and the atmosphere directly back into space, with the remaining portion striking the earth s surface. (This defines the earth s albedo as 0.31.) What is the intensity of the sun s radiation that strikes the surface of the earth? We will refer to this intensity as R IN (see figure 2). 7) Let s pause here for a second and look at the total power from the sun that hit s the earth. What is the total power of the sun s radiation that strikes the earth? (Remember that at any given time only one half of the spherical earth is in sun.) 4

5 8) The total energy use of the earth is around 1.6X10 13 Watts. How does the energy from the sun s radiation calculated above compare to this number?. Figure 2 Let s model the situation show in Figure 2 above. Energy from the Sun s thermal radiation (S) comes in, with 31% reflected back out. Since the earth is in energy balance, this same amount of energy must be sent back out again as thermal radiation from the earth. 9) Since the earth is in thermal balance, the power in from the sun s radiation must be re-radiated out by the earth. You now know how much radiation hits a square meter of the earth s surface from step (6) above (that s the in part). And in Step (3) you used a formula that tells out how much power the something will radiate if you know its temperature (that s the out part). Set your answer to Step (6) equal to the formula and work backwards to determine the temperature of the earth so that both sides are equal. (You can ignore the area part of the formula because we re doing everything per square meter.) 10) The temperature you calculated in step (9) is in the Kelvin scale. What is this temperature in o F? How does this temperature compare the temperature you see around you on an average day? 5

6 Part II One Layer Atmosphere Model Clearly something is missing in our model from Part I. We get a temperature that is clearly too low. But we haven t done any complex calculations. We have just used simple math and our basic thermodynamic equations. What can be wrong? Our problem is that we haven t included the warming effect to the earth of having an atmosphere, something that has been known for over a hundred years. Let s include the atmosphere in the simplest way possible. Let s assume it s just one layer. 1) We re going to use Wein s Law once again but this time let s figure out the peak wavelength for the radiation from the Earth. To get started, take the temperature of the Earth you calculated in Step (9) of Part I to calculate the peak wavelength from radiation Earth. How does it compare to the peak wavelength from the sun you calculated in Step 1 of Part II? Thermal Radiation IN from the Sun Thermal Radiation OUT from the Earth Figure 3 2) In figure 3 above, we see a plot of the real amount of radiation at each wavelength for both the sun (on the left) and the Earth (on the right). Mark on the plot the values for the peak wavelengths you calculated in Step 1 of Part I and Step 1 of Part II. (Notice the wavelengths ( ) on the graph are given in micrometers or 10-6 m.) How do they compare to the peaks of the curves shown? 6

7 In figure 3, the dark black vertical bands show wavelengths that cannot pass through our atmosphere. On the far left corner of the Sun s curve we see a black area indicating that the earth s atmosphere doesn t let radiation with the wavelength through. That s good because that blocked area represents ultra-violet radiation with high energy that can cause cancer. It s also good that we don t see any other dark lines because that means most of the Sun s thermal radiation gets through so we can see and to warm the earth. 3) On the other hand, on the right curve for the Earth s outgoing radiation, we see three dark lines, two of them very near the peak of the outgoing radiation. This means that the earth s atmosphere will not let that radiation out. What are the gases that cause those three dark lines? Figure 4 So now, we re going to calculate the Earth s temperature from the model shown in Figure 4 above. We have the same amount of thermal radiation coming in to the Earth s surface (R IN) from the sun, but now you have some additional radiation reflected back from the atmosphere (R A-DOWN). This gives us: R IN + R A-DOWN = R OUT (Eq 1) If we assume that the same amount of radiation is sent up from the atmosphere as is sent down, you have R A-UP = R A-DOWN (Eq 2) 7

8 Finally, we know that the energy from the radiation into the atmosphere must equal the energy from the radiation out. So we know R OUT = R A-UP + R A-DOWN (eq 3) 4) Take equations (2) and (3) do shown how R OUT and R A-DOWN are related. 5) Take this equation and substitute it into equation 1 to get a new relationship between R IN (from the Sun) and R OUT (from the Earth s surface). 6) Now take our formula for how thermal radiation from the Earth is related to the Earth s temperature (the one used in Steps (3) and (9) of Part I and calculate a new temperature for the earth. 7) Once again, this temperature is in Kelvin. What is this temperature in o C and o F? How does that compare to the temperature around you? The fact that the atmosphere stops some of the outgoing thermal radiation from the Earth and re-radiates it back towards the Earth is called the greenhouse effect. This effect has been known for over 100 years and is wellestablished in science. More importantly, we have shown that with a little high school math, basic thermodynamics, and a little guidance, we can roughly calculate the end result of the effect. We don t need a complex computer model to see the idea. We, of course, made some simplifying approximations, the most obvious of which is that the earth s atmosphere stops 100% of the outgoing radiation. We can see from figure 2 that since the black lines are near the peak of the earth s radiation, since might be close, but could use some work. If we used a bit more math and the power of Excel, we can make some additional adjustments, but we ll leave that for another class. 8